Karatsuba algorithm implementation: works for small ns, breaks for bigger ns

Question

I'm working on an implementation of the Karatsuba algorithm of multiplying numbers, but unlike most implementations using Strings as the primary data structure instead of BigNumbers or longs. I've written a recursive solution to the problem that appears to work for all n < 6, but for some reason it fails to work for odd ns greater than 6, despite all of the base cases working. Here's the karatsuba part of the program, with a few prints left behind from debugging. All of the methods used in this should work as intended, I tested them thoroughly. For a value factor1 = "180" and factor2 = "109", the correct result is outputted. For a value factor1 = "1111" and factor2 = "1111" the correct result is outputted. For a factor1 = "2348711" and factor2 = "8579294" the program outputs "20358060808034" when it should output "20150282190034". I've tried backtracing the logic, and I can't find where exactly it goes wrong. If anyone has any insight as to where something may not work, any help is appreciated.

public static String multiply(String factor1, String factor2) {
    // base case of length = 1
    System.out.println("Factor1 " + factor1 + " factor2 " + factor2);
    if (factor1.length() == 1 && factor2.length() == 1) {
        return smallNumberMultiplication(factor1, factor2);
    } else if (factor1.length() == 1 && factor2.length() == 2) { //these conditions needed for odd-size #s
        return smallNumberMultiplication(factor1, factor2); // max iteration = 10
    } else if (factor1.length() == 2 && factor2.length() == 1) {
        return smallNumberMultiplication(factor2, factor1); // max iteration = 10
    }

    // check which factor is smaller, find the index at which the value is split
    int numberLength = factor1.length();
    int middleIndex = numberLength / 2;
    // Find the power to which 10 is raised such that it follows Karatsuba's algorithm for ac
    int powerValue = numberLength + numberLength % 2;

    // divide both numbers into two parts bounded by middleIndex place
    String[] tempSplitString = splitString(factor1, middleIndex);
    String f1Large = tempSplitString[0], f1Small = tempSplitString[1];
    tempSplitString = splitString(factor2, middleIndex);
    String f2Large = tempSplitString[0], f2Small = tempSplitString[1];

    String multiplyHighestNumbers, multiplySmallestNumbers, multiplyMiddleNumbers;
    // large factor1 * large factor2
    multiplyHighestNumbers = multiply(f1Large, f2Large);
    // Multiply (f1Large + f1Small)*(f2Large + f2Small)
    multiplyMiddleNumbers = multiply(addTwoValues(f1Large, f1Small), addTwoValues(f2Large, f2Small));
    // small factor1 * small factor2
    multiplySmallestNumbers = multiply(f1Small, f2Small);

    // add trailing zeros to values (multiply by 10^powerValue)
    String finalHighestNumber = addTrailingZeros(multiplyHighestNumbers, powerValue);
    String finalMiddleNumber = addTrailingZeros(
            subtractTwoValues(subtractTwoValues(multiplyMiddleNumbers, multiplyHighestNumbers),
                    multiplySmallestNumbers),
            powerValue / 2);
    String finalSmallestNumber = multiplySmallestNumbers;

    // add each part together
    return removeLeadingZeros(addTwoValues(addTwoValues(finalHighestNumber, finalMiddleNumber), finalSmallestNumber));
}

Answer 1

I noticed two problems:

• using different values for splitting (middleIndex) and shifting (powerValue) (needlessly implemented by tacking on zeroes).
  For productHighParts("multiplyHighestNumbers") to be closer in length to the other products, use (factor1.length() + factor2.length()) / 4 (half the average length of both factors).
• this length has to be the length of the less significant part in splitString(), not the leading part.

(Note that the first two controlled statements can be combined:
if (factor1.length() <= 1 && factor2.length() <= 2).)